Classification of Non-Symmetric Rank-2 Hyperbolic Root Systems via Generalized Cartan Matrix H(a,b) — E8 Intelligence Research

FINDING: Classification of non-symmetric rank-2 hyperbolic root systems via generalized Cartan matrix H(a,b) with integer entries a,b satisfying ab ≥ 5, where a ≠ b yields distinct root length ratios sqrt(a/b). | MATH: Generalized Cartan matrix H(a,b) = [[2, -b], [-a, 2]]; a,b ∈ ℤ, ab ≥ 5; root length ratio = sqrt(a/b) for non-symmetric case (a ≠ b); symmetric case a = b yields equal lengths. | CONNECTION: Root length ratios sqrt(a/b) for integer a,b with ab ≥ 5 produce values such as sqrt(5/1)=2.236, sqrt(6/1)=2.449, sqrt(5/2)=1.581, etc. — none match the golden ratio 1.618 exactly but sqrt(5/2)=1.581 and sqrt(8/3)=1.633 approach it; the condition ab ≥ 5 excludes classical finite and affine root systems, placing these in hyperbolic (indefinite) domain. | DEPTH: 7 — This classification is a precise algebraic extension of Cartan-Killing theory into hyperbolic Kac-Moody algebras, with direct geometric interpretation via root length ratios and angles (determined by ab product). The non-sy Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com